A Decomposition Method for LQ Conditional McKean-Vlasov Control Problems with Random Coefficients

2026-04-13

AI Summary

This work addresses a class of linear-quadratic McKean–Vlasov control problems driven by two independent Wiener processes, featuring conditional expectations and random coefficients. The authors propose a novel decomposition framework that decouples the original problem into two subproblems: one with a constrained control set and the other unconstrained, both amenable to classical solution techniques. This approach circumvents the technical complexities and control restrictions typically arising from extended stochastic maximum principles or dynamic programming. The study establishes an equivalence between the well-posedness and solvability of the original problem and its decoupled counterparts. By leveraging variational methods, forward–backward stochastic differential equation (FBSDE) theory, and dynamic programming, the authors derive two explicit, decoupled linear FBSDE systems that fully characterize the optimal control structure, thereby reducing the intricate McKean–Vlasov problem to standard stochastic control settings.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Decentralized Web and Fediverse systems
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Machine-in-the-loop, human agency and autonomy

Abstract

We propose a decomposition method for solving a general class of linear-quadratic (LQ) McKean-Vlasov control problems involving conditional expectations and random coefficients, where the system dynamics are driven by two independent Wiener processes. Unlike existing approaches in the literature for these problems, such as the extended stochastic maximum principle and the extended dynamic programming methods, which often involve additional technical complexities and sometimes impose restrictive conditions on control inputs, our approach decomposes the original McKean-Vlasov control problem into two decoupled stochastic optimal control problems, one of which has a constrained admissible control set. These auxiliary problems can be solved using classical methods. We establish an equivalence between the well-posedness and solvability of the auxiliary problems and those of the original problem, and show that the sum of the optimal controls of the auxiliary problems yields the optimal control of the original problem. Moreover, by applying a variational method, we characterize the optimal solution to the McKean-Vlasov control problem via two decoupled sets of (non-McKean-Vlasov) linear forward-backward stochastic differential equations, each corresponding to one of the auxiliary problems. Finally, we show that standard dynamic programming can also be applied to solve the resulting auxiliary problems.

Problem

• McKean-Vlasov control
• linear-quadratic
• conditional expectation
• random coefficients
• stochastic optimal control

Innovation

• Decomposition method
• McKean-Vlasov control
• Linear-quadratic control
• Forward-backward SDEs
• Stochastic optimal control